Lowpass filters approximation based on modified Jacobi polynomials

The orthogonal Jacobi polynomials are not suitable for use as the characteristic function in the continuous- and discrete-time filter design, because they are not fulfilling the basic condition: to be pure odd or pure even. A simple modification of Jacobi polynomials, is performed to obtain a new filter approximating function is proposed. Magnitude frequency responses of obtained filters exhibit more general behaviour compared with that of classical Gegenbauer (ultraspherical) filters, due to one additional parameter available in the Jacobi polynomials. This parameter can be used to obtain magnitude response with either smaller passband ripple values (nearly monotonic behaviour), smaller group delay variations or sharper cutoff slope. The proposed modified Jacobi polynomials are not orthogonal, however, many known orthogonal polynomials can be obtained as theirs special cases.